posted on March 05, 2023, last updated on Saturday, November 23, 2024 at 10:51 AM

Support Vector Machines (SVM) are supervised learning models used for classification and regression tasks. The dual problem formulation of SVM provides an alternative perspective that can be computationally advantageous, particularly for high-dimensional feature spaces.

Primal Problem

The primal form of the SVM optimization problem is given by:

\[\begin{align} \min_{\mathbf{w}, b, \xi} \quad & \frac{1}{2} \|\mathbf{w}\|^2 + C \sum_{i=1}^{n} \xi_i \\ \text{subject to} \quad & y_i (\mathbf{w} \cdot \mathbf{x}_i + b) \geq 1 - \xi_i, \\ & \xi_i \geq 0, \quad i = 1, \ldots, n. \end{align}\]

Dual Problem

By applying the Lagrangian method, we can transform the primal problem into its dual form. The dual problem is expressed as:

\[\begin{align} \max_{\boldsymbol{\alpha}} \quad & \sum_{i=1}^{n} \alpha_i - \frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} \alpha_i \alpha_j y_i y_j (\mathbf{x}_i \cdot \mathbf{x}_j) \\ \text{subject to} \quad & \sum_{i=1}^{n} \alpha_i y_i = 0, \\ & 0 \leq \alpha_i \leq C, \quad i = 1, \ldots, n. \end{align}\]

Here, \( \boldsymbol{\alpha} = (\alpha_1, \alpha_2, \ldots, \alpha_n) \) are the Lagrange multipliers. The dual formulation simplifies the optimization problem by transforming it into a quadratic programming problem that only depends on the inner products of the training data points.

Advantages of the Dual Problem

1. Kernel Trick: The dual problem allows the use of kernel functions to handle non-linear decision boundaries without explicitly mapping the data to a higher-dimensional space.
2. Efficiency: For high-dimensional data, the dual problem can be more computationally efficient to solve than the primal problem.

In conclusion, the dual formulation of SVMs provides critical computational and conceptual benefits, particularly in handling complex, high-dimensional data.